I'm looking into mining, how it works, and how to get started. I see a "mining difficulty" from many sources and list a 3 billion something number without units.

What exactly does this number mean and how does it relate to the processing speed of the hardware used? For example, what predictions can be made on the number blocks you will solve with a 100 Gh/s ASIC at any given difficulty?

I assume Gh/s is giga hashes per second, though I don't know what a hash is either.

3 Answers
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(If I may repeat myself a bit...) Mining is like having a lot of people throwing weighted coins (such that 1 millionth of the time it comes up heads) and telling you when they hit a heads. If one such "heads" is reported every 10 minutes (600 seconds), you can make a very accurate estimation of how many times per second the coins are being flipped. In this example:

(1,000,000 flips/heads) / (600 seconds/heads) ~=
1,667 flips/second

The network difficulty is how you adjust this 1,000,000 figure so that the 600 figure stays consistent as the network's total hash power (1,667) changes.

When mining, your computer creates a block of data, which has a list of all of the transactions it knows about, includes a transaction that pays you the mining bonus, and then hashes that. If the hash happens to be a small enough number (as defined by the difficulty), the block is valid. If it's not, you increment a random number called a "nonce" that's in the block, so that the block has the same meaning but different data, and hashes the block again. Lather, rinse, repeat. (the nonce is included from the start, of course, but I didn't mention it to avoid being confusing)

Gigahashes per second, then, are how many billion times your hardware can do this hash per second.

If the pool you're in has 424 Th/s and you have 100 Gh/s, then you are about 1/4000th of your pool's total power. You can expect to be the one to solve about 1/4000th of the blocks (on average) that your pool finds.

(technical details intentionally omitted; those are available at my various links, and your link)

I'm looking into mining, how it works, and how to get started. I see a "mining difficulty" from many sources and list a 3 billion something number without units.

The difficulty is just an arbitrary value, it has no unit. A difficulty 8 block is 8 times more difficult than a difficulty. Today the difficulty is 3B, but this chances on a bi-weekly basis as the hashrate of the network varies.

For example, what predictions can be made on the number blocks you will solve with a 100 Gh/s ASIC at any given difficulty?

It takes about 4 Billion attempts to find a difficulty 1 block, you can scale up approximately from here. 100 GH means the hardware makes 100,000,000,000 attempts a second, or about a difficulty 1 block every 0.04 seconds. In practice a 100GH machine might find a block every 5 years or so, if you're lucky.

Pooled mining pools the resources to find blocks faster and split the income between the miners. It would average to about 0.13BTC a day at the moment, but it's dropping about 2% a day at the moment and doesn't look like stopping any time soon.

You won't make any income mining, nobody has except for the very first ASIC owners in January 2013.